American Options with Discontinuous Two-Level Caps

32 Pages Posted: 24 Jul 2017 Last revised: 31 Oct 2017

See all articles by Jerome Detemple

Jerome Detemple

Boston University - Department of Finance & Economics; Center for Interuniversity Research and Analysis on Organization (CIRANO)

Yerkin Kitapbayev

Boston University - Questrom School of Business

Date Written: October 30, 2017

Abstract

This paper examines the valuation of American capped call options with two-level caps. The structure of the immediate exercise region is significantly more complex than in the classical case with constant cap. When the cap grows over time, making extensive use of probabilistic arguments and local time, we show that the exercise region can be the union of two disconnected set. Alternatively, it can consist of two sets connected by a line. The problem then reduces to the characterization of the upper boundary of the first set, which is shown to satisfy a recursive integral equation. When the cap decreases over time, the boundary of the exercise region has piecewise constant segments alternating with non-increasing segments. General representation formulas for the option price, involving the exercise boundaries and the local time of the underlying price process, are derived. An efficient algorithm is developed and numerical results are provided.

Keywords: American capped option, optimal stopping, geometric Brownian motion, free-boundary problem, local time, integral equation

Suggested Citation

Detemple, Jerome and Kitapbayev, Yerkin, American Options with Discontinuous Two-Level Caps (October 30, 2017). Available at SSRN: https://ssrn.com/abstract=3005451 or http://dx.doi.org/10.2139/ssrn.3005451

Jerome Detemple

Boston University - Department of Finance & Economics ( email )

595 Commonwealth Avenue
Boston, MA 02215
United States
(617) 353-4297 (Phone)
(617) 353 6667 (Fax)

Center for Interuniversity Research and Analysis on Organization (CIRANO)

2020 rue University, 25th Floor
Montreal, Quebec H3C 3J7
Canada

Yerkin Kitapbayev (Contact Author)

Boston University - Questrom School of Business ( email )

595 Commonwealth Ave
Boston, MA 02466
United States

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